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Heegner cycles, modular forms and jacobi forms

Nils-Peter Skoruppa (1991)

Journal de théorie des nombres de Bordeaux

We give a geometric interpretation of an arithmetic rule to generate explicit formulas for the Fourier coefficients of elliptic modular forms and their associated Jacobi forms. We discuss applications of these formulas and derive as an example a criterion similar to Tunnel's criterion for a number to be a congruent number.

Heights and regulators of number fields and elliptic curves

Fabien Pazuki (2014)

Publications mathématiques de Besançon

We compare general inequalities between invariants of number fields and invariants of elliptic curves over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the elliptic curve side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of elliptic curves with dense rational points over a number field. This amounts to say that the arithmetic of CM fields...

Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze

Carlos D’Andrea, Teresa Krick, Martín Sombra (2013)

Annales scientifiques de l'École Normale Supérieure

We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion...

Hida families, p -adic heights, and derivatives

Trevor Arnold (2010)

Annales de l’institut Fourier

This paper concerns the arithmetic of certain p -adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a p -adic regulator, and the derivative of a p -adic L -function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first...

High rank eliptic curves of the form y2 = x3 + Bx.

Julián Aguirre, Fernando Castañeda, Juan Carlos Peral (2000)

Revista Matemática Complutense

Seven elliptic curves of the form y2 = x3 + B x and having rank at least 8 are presented. To find them we use the double descent method of Tate. In particular we prove that the curve with B = 14752493461692 has rank exactly 8.

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